Symplectic mean curvature flows in K\"ahler surfaces with positive holomorphic sectional curvatures

TL;DR

This paper investigates the behavior of symplectic mean curvature flows in K"ahler surfaces with positive holomorphic sectional curvatures, establishing conditions under which certain geometric quantities are preserved during the flow.

Contribution

It proves that if the ratio of maximum to minimum holomorphic sectional curvatures is less than 2, then a lower bound on  is maintained along the flow, advancing understanding of curvature flow stability.

Findings

Preservation of  lower bound under specific curvature ratio

Condition for stability of symplectic mean curvature flow

Relation between curvature ratio and flow behavior

Abstract

In this paper, we mainly study the mean curvature flow in K\"ahler surfaces with positive holomorphic sectional curvatures. We prove that if the ratio of the maximum and the minimum of the holomorphic sectional curvatures is less than 2, then there exists a positive constant $\delta$ depending on the ratio such that $\cos\alpha\geq\delta$ is preserved along the flow.